Modeling and Control of a 3PRS Robot Using Lagrange Method

Kinematic and Kinetic Modeling of a 3PRS mechanism is performed in this paper and its related controller is designed according to Computed Torque Method (CTM). This mechanism is a kind of spatial parallel robot with 6 DOFs, which is controlled using three active prismatic joints and three passive revolute joints and thus the system is categorized as constrained mechanism. Kinematic modeling of the robot is performed using Homogenous transformation matrix and its related Jacobian matrix is exracted. Therefore the position and velocity relation between the joint space and workspace are provided both in its direct and inverse modes. Dynamic equation of the system is also derived using Lagrange equation in the joint space of the robot and considering Lagrange multipliers in order to engage the related constraints of the system. Thus both of forward and inverse kinetic of the system are solved for the presented system. Finally, by the aid of the extracted dynamic equation, the robot is controlled using Computed Torque Method (CTM) equipped by a PD controller for compensatingthe uncertainties and disturbances. All of the modeling are verified by simulating the equations in MATLAB and comparing the direct and inverse modes. It is shown that using the extracted modeling and the designed controller for this parallel robot, the desired path can be easily tracked within the workspace of the robot.